### Abstract

The k-core of a graph is the largest subgraph with minimum degree at least k. For the Erdos-Rényi random graph G(n, m) on n vertives, with m edges, it is known that a giant 2-core grows simultaneously with a giant component, that is, when m is close to n/2. We show that for k ≥ 3, with high probability, a giant k-core appears suddenly when m reaches c_{k}n/2; here c_{k} = min_{λ>0} λ/π_{k}(λ) and π_{k}(λ) = P{Poisson(λ)≥k-1}. In particular, c_{3}≈3.35. We also demonstrate that, unlike the 2-core, when a k-core appears for the first time it is very likely to be giant, of size ≈p_{k}(λ_{k})n. Here λ_{k} is the minimum point of λ/π_{k}(λ) and p_{k}(λ_{k}) = P{Poisson(λ_{k})≥k}. For k = 3, for instance, the newborn 3-core contains about 0.27n vertices. Our proofs are based on the probabilistic analysis of an edge deletion algorithm that always find a k-core if the graph has one.

Original language | English (US) |
---|---|

Pages (from-to) | 111-151 |

Number of pages | 41 |

Journal | Journal of Combinatorial Theory, Series B |

Volume | 67 |

Issue number | 1 |

DOIs | |

State | Published - May 1996 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory, Series B*,

*67*(1), 111-151. https://doi.org/10.1006/jctb.1996.0036

**Sudden emergence of a giant k-core in a random graph.** / Pittel, Boris; Spencer, Joel; Wormald, Nicholas.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory, Series B*, vol. 67, no. 1, pp. 111-151. https://doi.org/10.1006/jctb.1996.0036

}

TY - JOUR

T1 - Sudden emergence of a giant k-core in a random graph

AU - Pittel, Boris

AU - Spencer, Joel

AU - Wormald, Nicholas

PY - 1996/5

Y1 - 1996/5

N2 - The k-core of a graph is the largest subgraph with minimum degree at least k. For the Erdos-Rényi random graph G(n, m) on n vertives, with m edges, it is known that a giant 2-core grows simultaneously with a giant component, that is, when m is close to n/2. We show that for k ≥ 3, with high probability, a giant k-core appears suddenly when m reaches ckn/2; here ck = minλ>0 λ/πk(λ) and πk(λ) = P{Poisson(λ)≥k-1}. In particular, c3≈3.35. We also demonstrate that, unlike the 2-core, when a k-core appears for the first time it is very likely to be giant, of size ≈pk(λk)n. Here λk is the minimum point of λ/πk(λ) and pk(λk) = P{Poisson(λk)≥k}. For k = 3, for instance, the newborn 3-core contains about 0.27n vertices. Our proofs are based on the probabilistic analysis of an edge deletion algorithm that always find a k-core if the graph has one.

AB - The k-core of a graph is the largest subgraph with minimum degree at least k. For the Erdos-Rényi random graph G(n, m) on n vertives, with m edges, it is known that a giant 2-core grows simultaneously with a giant component, that is, when m is close to n/2. We show that for k ≥ 3, with high probability, a giant k-core appears suddenly when m reaches ckn/2; here ck = minλ>0 λ/πk(λ) and πk(λ) = P{Poisson(λ)≥k-1}. In particular, c3≈3.35. We also demonstrate that, unlike the 2-core, when a k-core appears for the first time it is very likely to be giant, of size ≈pk(λk)n. Here λk is the minimum point of λ/πk(λ) and pk(λk) = P{Poisson(λk)≥k}. For k = 3, for instance, the newborn 3-core contains about 0.27n vertices. Our proofs are based on the probabilistic analysis of an edge deletion algorithm that always find a k-core if the graph has one.

UR - http://www.scopus.com/inward/record.url?scp=0030144334&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030144334&partnerID=8YFLogxK

U2 - 10.1006/jctb.1996.0036

DO - 10.1006/jctb.1996.0036

M3 - Article

AN - SCOPUS:0030144334

VL - 67

SP - 111

EP - 151

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 1

ER -